Sweep the action space before tuning difficulty — "hard" and "impossible" look identical from inside the game

Fact

When success depends on a continuous control input — an aim angle, a launch power, a steering curve — you can measure how hard the game actually is by brute-forcing that input from real positions and counting what scores. Two numbers come out of it:

• the fraction of positions that have any winning input at all, and
• the fraction of the input space that wins, when one exists.

Both are cheap to get if the simulation can be replayed headlessly, and neither is visible from playing.

Why it matters

A physically faithful carom table with heavy cloth and dead cushions gave, under a sweep of 3600 angles by 4 powers, no scoring stroke at all in nine of twenty positions, and about 0.05% of strokes elsewhere. From the inside that reads as a weak opponent and unfair aiming aids, and it invites tuning the search, the guide lines, and the difficulty curve — none of which can reach a position with no solution.

Livelier cushions, faster cloth and a slightly smaller table took it to twenty of twenty solvable. The same unchanged search code went from 20% to 50% success. The knob that mattered was in the physics; every knob reached for first was in the agent.

How to apply

• Sample finer than the success window you expect. If a sweep at 0.1° finds isolated hits, the window is near that width and a coarser sweep would have reported zero.
• Sweep from positions the game actually reaches — drive them forward with an existing agent between samples — not only from the opening state.
• Report solvable-position rate separately from win-rate. They fail differently: the first is a design fault, the second a tuning one.
• Near-zero density means change the world, not the search.
• Keep the sweep as a script and re-run it after any physics change; it is the only thing that distinguishes a hard game from a broken one.